There is at least one nice property of the Fibonacci numbers that depend on the indexing. The Fibonacci numbers form a "divisibility sequence" - that is, if $m | n$ then $F_m | F_n$. (If $m = 3$, for example, this is the fact that if $n$ is divisible by 3 then $F_n$ is even.) This doesn't hold if the Fibonacci sequence is indexed differently.

A lot of the relationships between Lucas and Fibonacci numbers, in turn, seem to be most cleanly expressed if the Lucas numbers are indexed as they are, for example $L_n^2 = 5 F_n^2 + 4 (-1)^n$.

If you're concerned about deriving these from the eigenvalues, though, I don't think any of this matters.